A Classical Search Game in Discrete Locations

Abstract

Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among n discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location i takes ti time units and detects the hider—if hidden there—independently with probability [Formula: see text] for [Formula: see text]. The hider aims to maximize the expected time until detection, whereas the searcher aims to minimize it. We prove the existence of an optimal strategy for each player. In particular, any optimal mixed hiding strategy hides in each location with a nonzero probability, and there exists an optimal mixed search strategy that can be constructed with up to n simple search sequences.

Counterintuitive prey strategies against predators with finite budgets: protection heterogeneity among sites matters more than their number

“…Once m gets closer to the deadline, such risks may no longer be needed, and the prey would hide in the locations with the lowest capture probabilities. Somewhat similar considerations of opposite dual motivations are at the heart of [27] about searching with deadlines. However, in our work, the predator must keep an eye on its budget, and it does not need to find the prey as soon as possible.…”

“…A prey acting in a minimax point of view would try to waste the predator's budget m, even if it implies hiding in the locations with the highest capture probabilities (location n), as it is important to make big decreases in m. Once m gets closer to the deadline, such risks may no longer be needed, and the prey would hide in the locations with the lowest capture probabilities. Somewhat similar considerations of opposite dual motivations are at the heart of [27] about searching with deadlines. However, in our work, the predator must keep an eye on its budget, and it does not need to find the prey as soon as possible.…”